L(s) = 1 | + 0.104·2-s − 3-s − 1.98·4-s − 0.431·5-s − 0.104·6-s + 2.76·7-s − 0.415·8-s + 9-s − 0.0449·10-s − 1.08·11-s + 1.98·12-s − 2.81·13-s + 0.288·14-s + 0.431·15-s + 3.93·16-s + 5.82·17-s + 0.104·18-s − 7.25·19-s + 0.859·20-s − 2.76·21-s − 0.113·22-s − 3.48·23-s + 0.415·24-s − 4.81·25-s − 0.292·26-s − 27-s − 5.50·28-s + ⋯ |
L(s) = 1 | + 0.0736·2-s − 0.577·3-s − 0.994·4-s − 0.193·5-s − 0.0425·6-s + 1.04·7-s − 0.146·8-s + 0.333·9-s − 0.0142·10-s − 0.328·11-s + 0.574·12-s − 0.779·13-s + 0.0769·14-s + 0.111·15-s + 0.983·16-s + 1.41·17-s + 0.0245·18-s − 1.66·19-s + 0.192·20-s − 0.603·21-s − 0.0241·22-s − 0.727·23-s + 0.0847·24-s − 0.962·25-s − 0.0574·26-s − 0.192·27-s − 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 717 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 239 | \( 1 + T \) |
good | 2 | \( 1 - 0.104T + 2T^{2} \) |
| 5 | \( 1 + 0.431T + 5T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 - 5.82T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 + 3.48T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 41 | \( 1 + 5.40T + 41T^{2} \) |
| 43 | \( 1 + 5.91T + 43T^{2} \) |
| 47 | \( 1 + 0.795T + 47T^{2} \) |
| 53 | \( 1 + 3.47T + 53T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 3.98T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 1.10T + 73T^{2} \) |
| 79 | \( 1 + 0.409T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.06743837153485138964750747308, −9.195714409888173475080919364214, −8.004527257282949457433496745818, −7.75404619173823748658813961129, −6.20893250150638808364343812371, −5.27168320251255361708567759951, −4.63122618280757220999159820337, −3.64813542419445510159247722876, −1.80409239528569592555048472095, 0,
1.80409239528569592555048472095, 3.64813542419445510159247722876, 4.63122618280757220999159820337, 5.27168320251255361708567759951, 6.20893250150638808364343812371, 7.75404619173823748658813961129, 8.004527257282949457433496745818, 9.195714409888173475080919364214, 10.06743837153485138964750747308